×
Ben Cheikh, Y.; Lamiri, I.; Ouni, A.

On Askey-scheme and \(d\)-orthogonality. I: A characterization theorem. (English) Zbl 1188.33018

J. Comput. Appl. Math. 233, No. 3, 621-629 (2009).
Authors’ abstract: This is the first in a series of papers dealing with generalized hypergeometric \(d\)-orthogonal polynomials extending the polynomial families in the Askey-scheme. In this paper, we give a characterization theorem to introduce new examples of generalized hypergeometric \(d\)-orthogonal polynomials to be studied in the forthcoming works. For \(d=1\), we obtain a unification of some known characterization theorems in the orthogonal polynomials theory.
Reviewer: Khodabakhsh Hessami Pilehrood (Shahrekord)

Cited in 1 Review
Cited in 16 Documents

MSC:

33C47 Other special orthogonal polynomials and functions
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
33C20 Generalized hypergeometric series, \({}_pF_q\)

Keywords:

\(d\)-orthogonality; generalized hypergeometric polynomials; Askey-scheme; Gould-Hooper polynomials; Humbert polynomials
Cite Review PDF
Full Text: DOI

References:

[1] Luke, Y. L., The Special Functions and Their Approximations, vol. I (1969), Academic Press: Academic Press New York, San Francisco, London · Zbl 0193.01701
[2] R. Koekoek, R.F. Swarttouw, The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue, No. 98-17, Faculty of the Technical Mathematics and Informatics, Delft University of Technology, Delft, 1998; R. Koekoek, R.F. Swarttouw, The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue, No. 98-17, Faculty of the Technical Mathematics and Informatics, Delft University of Technology, Delft, 1998
[3] Àlvarez-Nodarse, R.; Costas-Santos, R. S., Limit relations between \(q\)-Krall type orthogonal polynomials, J. Math. Anal. Appl., 322, 158-176 (2006) · Zbl 1096.33011
[4] Beckermann, B.; Coussement, J.; Van Assche, W., Multiple Wilson and Jacobi-Piñeiro polynomials, J. Approx. Theory, 132, 155-181 (2005) · Zbl 1066.33005
[5] Van Iseghem, J., Vector orthogonal relations. Vector QD-algorithm, J. Comput. Appl. Math., 19, 141-150 (1987) · Zbl 0626.65013
[6] Maroni, P., L’orthogonalité et les récurrences de polynômes d’ordre supérieur à deux, Ann. Fac. Sci. Toulouse, 10, 105-139 (1989) · Zbl 0707.42019
[7] Shohat, J., Sur les polynômes orthogonaux généralisés, C. R. Acad. Sci., 207, 556-558 (1938) · Zbl 0019.40503
[8] Chihara, T. S., An Introduction to Orthogonal Polynomials (1978), Gordan and Breach: Gordan and Breach New York, London, Paris · Zbl 0389.33008
[9] Aptekarev, A. I., Multiple orthogonal polynomials, J. Comput. Appl. Math., 99, 423-447 (1998) · Zbl 0958.42015
[10] Aptekarev, A. I.; Branquinho, A.; Van Assche, W., Multiple orthogonal polynomials for classical weights, Trans. Amer. Math. Soc., 355, 3887-3914 (2003) · Zbl 1033.33002
[11] Arvesú, J.; Coussement, J.; Van Assche, W., Some discrete multiple orthogonal polynomials, J. Comput. Appl. Math., 153, 19-45 (2003) · Zbl 1021.33006
[12] Aptekarev, A. I.; Marcellán, F.; Rocha, I. A., Semi classical multiple orthogonal polynomials and the properties of Jacobi-Bessel polynomials, J. Approx. Theory, 90, 117-146 (1997) · Zbl 0878.33004
[13] De Bruin, M. G., Simultaneous Padé approximation and orthogonality, Lecture Notes in Math., 1171, 74-83 (1985) · Zbl 0587.41013
[14] Van Assche, W., Multiple Orthogonal Polynomials, Continued Functions, vol. 236 (1984), American Math Soc. Prov.
[15] Van Assche, W.; Coussement, E., Some classical multiple orthogonal polynomials, J. Comput. Appl. Math., 127, 317-347 (2001) · Zbl 0969.33005
[16] Van Assche, W.; Yakubovich, S. B., Multiple orthogonal polynomials associated with MacDonald functions, Integral Transforms Spec. Funct., 9, 229-244 (2000) · Zbl 0959.42016
[17] Ben Cheikh, Y.; Ouni, A., Some generalized hypergeometric \(d\)-orthogonal polynomial sets, J. Math. Anal. Appl., 343, 464-478 (2008) · Zbl 1140.33003
[18] Ben Cheikh, Y.; Zaghouani, A., Some discrete d-orthogonal polynomial sets, J. Comput. Appl. Math., 156, 253-263 (2003) · Zbl 1055.33005
[19] Ben Cheikh, Y.; Zaghouani, A., \(d\)-orthogonality via generating functions, J. Comput. Appl. Math., 199, 2-22 (2007) · Zbl 1119.42009
[20] Ben Cheikh, Y.; Douak, K., A generalized hypergeometric d-orthogonal polynomial set, C. R. Acad. Sci. Paris, 331, 349-354 (2000) · Zbl 1002.33003
[21] Ben Cheikh, Y.; Douak, K., On the classical d-orthogonal polynomials defined by certain generating functions I, Bull. Belg. Math. Soc., 7, 107-124 (2000) · Zbl 0945.33007
[22] Ben Cheikh, Y.; Douak, K., On the classical d-orthogonal polynomials defined by certain generating functions II, Bull. Belg. Math. Soc., 8, 591-605 (2001) · Zbl 1036.33006
[23] Ben Cheikh, Y.; Gaied, M., Dunkl-Appell d-orthogonal polynomials, Integral Transforms Spec. Funct., 18, 581-597 (2007) · Zbl 1137.42005
[24] Ben Cheikh, Y.; Ben Romdhane, N., d-orthogonal Polynomial Sets of Tchebytchev Type, World Scientific, 100-111 (2005) · Zbl 1221.33014
[25] Ben Romdhane, N., d-orthogonal Faber polynomials, Integral Transforms Spec. Funct., 18, 663-677 (2007) · Zbl 1126.42006
[26] Douak, K., The relation of the d-orthogonal polynomials to the Appell polynomials, J. Comput. Appl. Math., 70, 279-295 (1996) · Zbl 0863.33007
[27] Douak, K.; Maroni, P., On d-orthogonal Tchebyshev polynomials I, Appl. Numer. Math., 24, 23-53 (1997) · Zbl 0881.33024
[28] Douak, K.; Maroni, P., On d-orthogonal Tchebyshev polynomials II, Methods Appl. Anal., 4, 404-429 (1997) · Zbl 0904.33003
[29] Zaghouani, A., Some basic d-orthogonal polynomial sets, Georgian Math. J., 12, 583-593 (2005) · Zbl 1091.42020
[30] Gould, H. W.; Hopper, A. T., Operational formulas connected with two generalizations of hermite polynomials, Duke Math. J., 29, 51-63 (1962) · Zbl 0108.06504
[31] Humbert, P., Some extentions of Pincherle’s polynomials, Proc. Edinburgh Math. Soc., 39, 21-24 (1920)
[32] Rainville, E. D., Special Functions (1960), The Macmillan Company: The Macmillan Company New York · Zbl 0050.07401
[33] Lamiri, I.; Ouni, A., \(d\)-orthogonality of Hermite type polynomials, Appl. Math. Comput., 202, 24-43 (2008) · Zbl 1154.33004
[34] Lamiri, I.; Ouni, A., \(d\)-orthogonality of Humbert and Jacobi type polynomials, J. Math. Anal. Appl., 341, 24-51 (2008) · Zbl 1218.33011
[35] Ben Cheikh, Y.; Douak, K., On two-orthogonal polynomials related to Bateman’s \(J_n^{u, v}\)-function, Methods Appl. Anal., 7, 641-662 (2000) · Zbl 1009.33014
[36] Abdul-Halim, N. A.; Al-Salam, W. A., A characterization of Laguerre polynomials, Rend. del Seminario Mat. Univ. Padova. Math., 34, 176-197 (1964) · Zbl 0124.03504
[37] Al-Salam, N., Orthogonal polynomials of hypergeometric type, Ser. Math. Inform., 33, 109-121 (1966) · Zbl 0141.07201
[38] Routh, E., On some properties of certain solutions of a differential equation of the second order, Proc. London Math. Soc., 16, 245-261 (1884) · JFM 17.0315.02
[39] Masjedjamei, M., Three finite classes of hypergeometric orthogonal polynomials and their application in functions approximation, Integral Transforms Spec. Funct., 13, 169-190 (2002) · Zbl 1017.33005
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.